Ray solvable linear systems and ray S2NS matrices

AbstractRay solvable linear systems and ray S2NS matrices are complex generalizations of the sign solvable linear systems and S2NS matrices. We use the determinantal ray unique matrices (instead of ray nonsingular matrices) as a generalization of SNS matrices, to generalize some fundamental results of S2NS matrices from the real case to complex case, such as the graph theoretical characterization, the inverse ray patterns and the upper bound of the number of nonzero entries of S2NS matrices. The well known characterization of the sign solvable linear systems (in terms of the L-matrices and S∗ matrices) is also generalized to ray solvable linear systems, and the relationships between the ray S∗-matrices and real S∗-matrices are investigated. Some examples are also given to illustrate that some results, such as the characterization of the sign inconsistent linear systems, do not carry over to the complex case

Extremal properties of raynonsingular matrices

A ray–nonsingular matrix is a square complex matrix, A, such that each complex matrix whose entries have the same arguments as the corresponding entries of A is nonsingular. Extremal properties of ray– nonsingular matrices are studied in this paper. Combinatorial and probabilistic arguments are used to prove that if the order of a ray– nonsingular matrix is at least 6, then it must contain a zero entry, and that if each of its rows and columns have an equal number, k, of nonzeros, then k ≤ 13. ∗ This paper was written while Professor Lee was visiting the University of Wyoming and was supported by a 1996 Post-doctoral Fellowship from KOSEF